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How do we physically choose a co-ordinate system for making astronomical observations?

In a special relativistic system, the definition of relative velocity, clock synchronization is well understood and can be measured between observers located at arbitrarily separated points (this has been discussed in an earlier post). However, for a curved space-time, this construction is only true locally. Say, if two observers are far away from one another and they want to synchronize their clocks, they would need to measure the distance between them and also their relative velocities. The only unambiguous way to measure relative velocity between two observers is when their velocity vectors are compared on the same tangent plane. If these observers are at different locations, then there is no unique way one can define their relative velocity (discussed in this post). Further, to determine the distance between these observers, one would ideally need to know the metric (at least, in the neighborhood containing the two observers). This can be determined to some extent from indirect observations. For example, in a cosmological setting, one could cook up a metric in the vicinity of observer's past light cone from independent astronomical observables, like observed redshift, number count density, luminosity etc (see Ellis et al., 1985). However, these observables do not completely determine all of the metric components (but they completely determine the FRW geometry).

Given these uncertainties in the definition of relative velocities and metric of our observable surrounding, are there ways by which one could physically construct a co-ordinate system? If yes, are these co-ordinate system reliable ? And to what extent (length scale)?

KP99
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  • Modern astronomy uses the ICRF. Of course, you also need a relativity-aware time scale; see https://en.wikipedia.org/wiki/Barycentric_Coordinate_Time & https://en.wikipedia.org/wiki/Barycentric_Dynamical_Time And see The JPL Planetary and Lunar Ephemerides DE440 and DE441 by Park et al for details on precisely converting Earth-based data using UTC to a form suitable for precise ephemeris calculations. – PM 2Ring Dec 22 '22 at 13:01
  • @PM2Ring Thank you for the links!! – KP99 Dec 22 '22 at 13:05
  • @PM2Ring Just a follow up question on ICRF: I realized that the standard convention (in the solar system scale) is to use the simple spherical coordinates (like in Newtonian gravity) and consider relativistic corrections as post Newtonian approximation. Do we also use this convention of using spherical coordinates all the way upto cosmological scale? In theory, one can define arbitrary co-ordinate which mimics , say, radial coordinate at solar system and cosmological scale but deviate elsewhere. How do we know that we aren't using such arbitrary coordinate while making observations? – KP99 Jan 01 '23 at 18:50
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    Spherical coords are great for observations. We can measure directions on the celestial sphere to high precision, but our knowledge of the distances to bodies outside the Solar System is much less precise. Hipparcos & Gaia have greatly improved the situation for our galactic neighbourhood, but there are still many important nearby stars whose distances are only poorly known, eg Betelgeuse. – PM 2Ring Jan 01 '23 at 20:46
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    Clearly, it's pointless to worry about relativistic corrections to such crude measurements. Inside the Solar System, spherical coords are convenient for recording observations and pointing telescopes, but for orbit calculations it's more convenient to use cartesian coords. JPL Horizons uses XYZ coords, essentially in the ICRF frame, but (of course) it can also supply data in various equatorial or ecliptic spherical coords. I have some demos here. – PM 2Ring Jan 01 '23 at 20:54
  • Thank you for the explanation. Just one last question: In this standard convention, do we always treat relativistic corrections as perturbative corrections (then it makes sense to ignore these corrections for crude measurement)? Say, in stellar parallax method, if the curvature b/w sun and and a distant star is sufficiently curved, then the light rays can deviate from being a straight line giving rise to a different apparent position of the star. In this case, do we completely ignore these curvature/relativistic effects while making measurements? – KP99 Jan 02 '23 at 15:20
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    Curvature of light rays inside the galaxy is mostly negligible, unless the ray happens to pass very close to a very dense body (a white dwarf, neutron star, or black hole), but even when that happens you need to know the mass of the body to calculate the deflection. When looking at distant galaxies, curvature is very important, and many of our most distant observations rely on gravitational lensing. We have a few questions about lensing on Astronomy.SE. – PM 2Ring Jan 02 '23 at 15:34

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